Rigid supersymmetric theories in 4d Riemannian space

TL;DR

This work develops a comprehensive framework for rigid $\mathcal{N}=1$ supersymmetry on 4D Euclidean manifolds by constructing the Lagrangian directly in Euclidean signature and encoding supersymmetry in Killing-spinor equations on backgrounds defined by $(g_{mn}, b_m, M, \bar{M})$. It reframes the Killing-spinor problem in terms of torsion-class constraints of local $G$-structures (trivial or $SU(2)$), providing explicit, necessary-and-sufficient conditions for unbroken supersymmetry and a practical procedure to verify backgrounds via frame derivatives. The paper also analyzes a broad set of explicit backgrounds—K3, products $T^d\times S^{4-d}$, $T^d\times H^{4-d}$, $T^2\times \mathcal{M}_2$, and conformally flat spaces—showing how Weyl curvature and boundary terms affect the amount of preserved supersymmetry (ranging from $\mathcal{N}=1$ to $\mathcal{N}=4$ in special cases). These results extend the catalogue of rigid supersymmetric backgrounds beyond previously known conformally flat cases and provide a robust toolkit for discussions of localization and sigma-model geometry in curved Euclidean spaces.

Abstract

We consider rigid supersymmetric theories in four-dimensional Riemannian spin manifolds. We build the Lagrangian directly in Euclidean signature from the outset, keeping track of potential boundary terms. We reformulate the conditions for supersymmetry as a set of conditions on the torsion classes of a suitable SU(2) or trivial G-structure. We illustrate the formalism with a number of examples including supersymmetric backgrounds with non-vanishing Weyl tensor.